Optimal Quantum Algorithm for Estimating Fidelity to a Pure State

TL;DR

The paper addresses estimating the fidelity between a mixed state $\rho$ and a pure state $|\psi\rangle$, defined as $F(\rho,|\psi\rangle)=\sqrt{\langle\psi|\rho|\psi\rangle}$, under purified quantum query access. The authors introduce an optimal quantum algorithm that achieves $O(1/\varepsilon)$ queries to the state-preparation oracles, encoding the fidelity into amplitudes of an efficiently preparable state and applying square-root amplitude estimation; this yields a quadratic improvement over the folklore $O(1/\varepsilon^2)$ approach. The method generalizes to two mixed states by encoding $\sqrt{\operatorname{tr}(\rho\sigma^2)}$, recovering $F$ when $\sigma$ is pure, and it is proven to be optimal via a $\Omega(1/\varepsilon)$ lower bound that extends to arbitrary rank. The results unify and extend purity-based fidelity estimation, provide a generalized framework, and suggest further directions, such as encoding $\sqrt{\operatorname{tr}(\rho\sigma)}$ and connections to Frobenius norms.

Abstract

We present an optimal quantum algorithm for fidelity estimation between two quantum states when one of them is pure. In particular, the (square root) fidelity of a mixed state to a pure state can be estimated to within additive error $\varepsilon$ by using $Θ(1/\varepsilon)$ queries to their state-preparation circuits, achieving a quadratic speedup over the folklore $O(1/\varepsilon^2)$. Our approach is technically simple, and can moreover estimate the quantity $\sqrt{\operatorname{tr}(ρσ^2)}$ that is not common in the literature. To the best of our knowledge, this is the first query-optimal approach to fidelity estimation involving mixed states.

Figures (4)

• Figure 1: The SWAP test for estimating $\mathrm{F}\lparen\rho, \lvert\psi\rangle\rparen$.
• Figure 2: The quantum circuit for estimating $\mathrm{F}\lparen\lvert\varphi\rangle, \lvert\psi\rangle\rparen$.
• Figure 3: The encoding unitary operator $W$ for $\mathrm{F}\lparen\rho, \lvert\psi\rangle\rparen$.
• Figure 4: Restructured circuit of \ref{['fig:q-von-ex']}, illustrating the operator $\sigma_{\mathsf{A}}\otimes I_{\mathsf{B}}$ and its action on the purification $\lvert\rho\rangle_{\mathsf{AB}}$.

Theorems & Definitions (16)

• Theorem 1.1: Estimating fidelity to a pure state, \ref{['thm:fidelity_to_pure']} simplified
• Definition 2.1: Purified quantum query access
• Theorem 3.1: Quantum amplitude estimation, BHMT02
• Theorem 3.2: Quantum square root amplitude estimation, Wan24
• Proposition 4.1
• proof
• Theorem 4.2: Estimating fidelity to a pure state
• proof
• Proposition 4.3
• proof